For the whole points and to try things there's https://learn.chef.io which allow you to test automate also.
Mainly chef automate is the next iteration bringing together 4 commercial products from Chef: chef manage (UI), chef reporting, chef compliance and delivery (CI/CD) with the addition of push jobs which was open sourced a little before.
Inspec is another new tool for compliance, it is based on serverspec but not extending it not using its code base.
Automate is not mandatory to author cookbooks at all, it is just a convenient way to display back status of nodes either on configuration point of view or compliance point of view (see the audit cookbook for inspec usage within a run), you can still does everything on your nodes on the free open source model, you'll have to handle the feedback loop yourself, either through a handler or any other way of your choice as it was already the case.
The CI/CD part of Automate doesn't replace any tool, it's more another approach to CI/CD pipelines, so you may use it to do the usual linting (foodcritic/cookstyle), test (chefspec), and then upload to your chef-server for use. It can handle deploys through the push jobs feature and after review and approval of changes.
Presentation of automate and the related documentation (workflow is the first concept, others are accessible from the menu on the left)
